Unbounded continuous operators and unbounded Banach–Saks property in Banach lattices

Abstract

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence (uaw-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded continuous operators between Banach lattices which maps uaw-Cauchy sequences to weakly (uaw- or norm) convergent sequences. This allows us to characterize KB-spaces and reflexive spaces in terms of these operators, too. Furthermore, we consider the unbounded Banach–Saks property as an unbounded version of the weak Banach–Saks property. There are many relations between spaces possessing the unbounded Banach–Saks property with spaces fulfilled by different types of the known Banach–Saks property. In particular, we characterize order continuous Banach lattices in terms of these relations, as well.